We will use Griffiths description of the cohomology of hypersurfaces in [1]}.
Let \({\sf T }\)  be the parameter space of smooth hypersurfaces of degree \(d\)  in \(\mathbb {P}^{2n+2}\) , \(F=\sum _{t\in {\sf T }} t_{\alpha }x^\alpha \)  be the universal homogeneous polynomial of degree \(d\)  in \(x=(x_0,x_1,\cdots ,x_{2n+2})\)  and \({\sf X }:=\lbrace F=0\rbrace \subset \mathbb {P}^{2n+2}\times {\sf T }\) . We define
\(\omega _P={\rm Resi }\frac{P\Omega }{F^k},\ \ \deg (P)=k\cdot d-(2n+3),\ \ \Omega :=\sum _{i=0}^{2n+2}(-1)^{i}x_i\widehat{dx_i}\) 
